Marathe K. Topics in Physical Mathematics

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58 2 Topology

ferentiable manifolds. We now discuss the cohomology theory based on the

cochain complex of diﬀerential forms on a manifold. This is the well known

de Rham cohomology with real coeﬃcients.

2.5 de Rham Cohomology

The de Rham complex of an m-dimensional manifold M is the cochain

complex (Λ(M),d)givenby

0 −→ Λ

(M)

−→ Λ

(M)

−→···

−→ Λ

(M) −→ 0 . (2.6)

The cohomology H

∗

(Λ(M),d) is called the de Rham cohomology of M and

is denoted by H

∗

deR

(M). The de Rham cohomology has a natural structure of

graded algebra induced by the exterior product. The product on homogeneous

elements is given by the map

∪ : H

(M; P) ×H

(M; P) → H

i+j

(M; P)

deﬁned by

([α], [β]) → [α ∧β].

This induced product in cohomology is in fact a special case of a cohomology

operation called the cup product (see Spanier [357]).

If M,N are manifolds then we have

∗

deR

(M ×N)=H

∗

deR

(M)

⊗H

∗

deR

(N),

where

⊗ denotes the graded tensor products. In particular, we can express

the cohomology of M × N in terms of the cohomologies of M and N as

follows:

deR

(M ×N)=



k=i+j

deR

(M)⊗H

deR

(N). (2.7)

In fact, the above formula holds more generally and is called the K¨unneth

formula.

There is a canonical map ρ called the de Rham homomorphism

deR

(M) → (

∞

(M; R))



∼

∞

(M; R)

given by the following pairing between de Rham cohomology classes [α]and

real diﬀerentiable singular homology classes [c]

([α], [c]) →



α.

2.5 de Rham Cohomology 59

One can show that this map ρ is independent of the choice of α ∈ [α]and

c ∈ [c]. The classical de Rham theorem says that the map ρ is an isomorphism.

Thus, ∀q we have

deR

(M)

∼

∞

(M; R)

∼

(M; R).

Let (M,g) be an oriented, closed (i.e., compact and without boundary)

Riemannian manifold with metric volume form μ. Recall that the Hodge–

de Rham operator Δ = d ◦δ + δ ◦ d maps Λ

(M) → Λ

(M), ∀k and that

the Hodge star operator ∗ maps Λ

(M) → Λ

n−k

(M), ∀k. For further details,

see Chapter 3. The map

 ,  : Λ

(M) × Λ

(M) → R

deﬁned by

α, β =



α ∧∗β =



g(α, β)μ (2.8)

is an inner product on Λ

(M). One can show that, for σ ∈ Λ

k+1

(M), we have

dα, σ = α, δσ and Δα, β = α, Δβ. (2.9)

That is, δ is the adjoint of d and Δ is self-adjoint with respect to this inner

product. Furthermore, Δα =0ifandonlyifdα = δα =0. An element of the

set

:= {α ∈ Λ

(M) | Δα =0} (2.10)

is called a harmonic k-form. It follows that a k-form is harmonic if and only

if it is both closed and coclosed. The set H

is a subspace of Λ

(M). Using

these facts one can prove (see, for example, Warner [396]) the Hodge decom-

position theorem, which asserts that H

is ﬁnite-dimensional and Λ

(M)

has a direct sum decomposition into the orthogonal subspaces d(Λ

(M)),

δ(Λ

(M)), and H

. Thus any k-form α canbeexpressedbytheformula

α = dβ + δγ + θ, (2.11)

where β ∈ Λ

k−1

(M), γ ∈ Λ

k+1

(M)andθ is a harmonic k-form. For α in a

given cohomology class, the harmonic form θ of equation (2.11) is uniquely

determined. Thus, we have an isomorphism of the kth cohomology space

(M; R) with the space of harmonic k-forms H

. Therefore, the kth Betti

number b

is equal to the dim H

. This is an illustration of a relation between

physical or analytic data (the solution space of a partial diﬀerential operator)

on a manifold and its topology. A far-reaching, nonlinear generalization of

this idea relating the solution space of Yang–Mills instantons to the topology

of 4-manifolds appears in the work of Donaldson (see Chapter 9 for further

details).

60 2 Topology

2.5.1 The Intersection Form

Let M be a closed (i.e., compact, without boundary), connected, oriented

manifold of dimension 2n.Letv denote the volume form on M deﬁning the

orientation. We shall use the de Rham cohomology to deﬁne ι

,thein-

tersection form of M as follows. Let α, β ∈ Λ

(M) be two closed n-forms

representing the cohomology classes a, b ∈ H

(M; Z) ⊂ H

(M; R) respec-

tively, i.e., a =[α]andb =[β]. Now α∧β ∈ Λ

(M) and hence



α∧β is well

deﬁned with respect to the volume form v. It can be shown that this integral

is independent of the choice of forms α, β representing the cohomology classes

a, b and takes values that are integral multiples of the volume of M.Thus,

we can deﬁne the binary operator

: H

(M; Z) ×H

(M; Z) → Z

(a, b)=



(α ∧β).

In what follows we shall use the same letter to denote the cohomology class

and an n-form representing that class. It can be shown that ι

is a symmetric,

non-degenerate bilinear form on H

(M; Z). This symmetric, non-degenerate

form ι

is called the intersection form of M . The deﬁnition given above

works only for smooth manifolds. However, as is the case with de Rham co-

homology, the intersection form does not depend on the diﬀerential structure

and is a topological invariant. In particular, it is deﬁned for topological man-

ifolds. In fact, the intersection form can also be deﬁned for non-orientable

manifolds by considering cohomology or homology with coeﬃcients in Z

in-

stead of Z. Now for a compact manifold M, H

(M; Z) is a ﬁnitely generated

free Abelian group of rank b

(the nth Betti number), i.e., an integral lattice

of rank b

. Thus, the intersection form gives us the map

ι : M → ι

which associates to each compact, connected, oriented topological manifold

M of dimension 2n a symmetric, non-degenerate bilinear form ι

on a lattice

of rank b

.Let(b

−

) be the signature of the bilinear form ι

.Ifb

> 0and

, 1 ≤ j ≤ b

,isabasisofthelatticeH

(M; Z), then the intersection form

is completely determined by the matrix of integers ι

), 1 ≤ j, k ≤ b

From Poincar´e duality it follows that the intersection form is unimodular,

i.e.,

|det(ι

))| =1.

If b

= 0 then we take ι

:= ∅ the empty form. Recall that on the abstract

level two forms ι

,ι

on lattices L

, respectively, are said to be equivalent

if there exists an isomorphism of lattices f : L

→ L

such that f

∗

= ι

2.6 Topological Manifolds 61

The intersection form plays a fundamental role in Freedman’s classiﬁcation

of topological 4-manifolds. We give a brief discussion of this result in the next

section.

2.6 Topological Manifolds

In this section we discuss topological manifolds with special attention to

low-dimensional manifolds. The following theorem gives some results on the

existence of smooth structures on topological manifolds.

Theorem 2.7 Let M be a closed topological manifold of dimension n.Then

we have the following results:

1. For n ≤ 3 there is a unique compatible smooth structure on M.

2. For n = 4 there exist (inﬁnitely many) simply connected manifolds that

admit inﬁnitely many distinct smooth structures. It is not known whether

there are manifolds that admit only ﬁnitely many distinct smooth struc-

tures.

3. For n ≥ 5 there are at most ﬁnitely many distinct compatible smooth

structures.

Thus, dimension 4 seems very special. This is also true for open topologi-

cal manifolds. There is a unique smooth structure on R

, n = 4, compatible

with its standard topology. However, R

admits uncountably many smooth

structures. We do not know at this time if every open topological 4-manifold

admits uncountably many smooth structures. For further results in the sur-

prising world of 4-manifolds, see, for example, the book by Scorpan [343].

2.6.1 Topology of 2-Manifolds

The topology of 2-manifolds, or surfaces, was well known in the nineteenth

century. Smooth, compact, connected and oriented 2-manifolds are called

Riemann surfaces. An introduction to compact Riemann surfaces from

various points of view and their associated geometric structures may be found

in Jost [213]. They are classiﬁed by a single non-negative integer, the genus

g. The genus counts the number of holes in the surface. There is a standard

model Σ

for a surface of genus g obtained by attaching g handles to a

sphere (which has genus zero). Every smooth, compact, oriented surface is

diﬀeomorphic to one and only one Σ

. The classiﬁcation is sometimes given

in terms of the Euler characteristic of the surface. It is related to the genus

by the formula χ(Σ

)=2− 2g.

In the classical theory of surfaces, homology classes a, b ∈ H

(M; Z)were

represented by closed curves, which could be chosen to intersect transversally.

62 2 Topology

The intersection form was then deﬁned by counting the algebraic number

of intersections of these curves. Surfaces are completely classiﬁed by their

intersection forms. In the orientable case we have the following well known

result.

Theorem 2.8 Let M,N be two closed, connected, oriented surfaces. Then

∼

N (i.e., M is diﬀeomorphic to N ) if and only if the intersection forms

,ι

are equivalent. Moreover, if ι

= ∅ then M

∼

and if ι

= ∅ then

there exists k ∈ N such that ι

∼

kσ

, where





is a Pauli spin matrix, and kσ

is the block diagonal form with k entries of

,andM

∼

, where T

= S

× S

is the standard torus and kT

is the

connected sum of k copies of T

The Riemann surface together with a ﬁxed complex structure provides a

classical model for one-dimensional algebraic varieties or complex curves. A

compact surface corresponds to a projective curve. The genus of such a sur-

face is equal to the dimension of the space of holomorphic one forms on the

surface. This way of looking at a Riemann surface is crucial in the Gromov–

Witten theory. We will not consider it in this book. The genus has also a

topological interpretation as half the ﬁrst Betti number of the surface. We

note that the classiﬁcation of orientable surfaces given by the above theorem

can be extended to include non-orientable surfaces as well. We are inter-

ested in extending this theorem to the case of 4-manifolds. This was done by

Freedman in 1981. Before discussing his theorem we consider the topology of

3-manifolds where no intersection form is deﬁned.

2.6.2 Topology of 3-Manifolds

The classiﬁcation of manifolds of dimension 3 or higher is far more diﬃcult

than that of surfaces. It was initiated by Poincar´e in 1900. The year 1900 is

famous for the Paris ICM and Hilbert’s lecture on the major open problems

in mathematics. The classiﬁcation of 3-manifolds was not among Hilbert’s

problems. Armed with newly minted homology groups and his fundamental

group Poincar´e began his study by trying to characterize the simplest 3-

manifold, the sphere S

. His ﬁrst conjecture was the following:

Let M be a compact connected 3-manifold with the same homology groups

as the sphere S

.ThenM is homeomorphic to S

In attempting to prove this conjecture Poincar´e found a 3-manifold P

that provided a counterexample to the conjecture. This 3-manifold P is now

called the Poincar´e homology sphere. It is denoted by Σ(2, 3, 5) as it can

be represented as a special case of the Brieskorn homology 3-spheres,

2.6 Topological Manifolds 63

Σ(a

):={(z

) | z

+ z

=0}∩S

∈ N.

We will compute various new invariants of the Brieskorn homology 3-spheres

in Chapter 10. Poincar´e’s original ingenious construction of P can be de-

scribed via well known geometric ﬁgures. It is the space of all regular icosa-

hedra inscribed in the standard unit 2-sphere. Note that each icosahedron is

uniquely determined by giving one vertex on the sphere (2 parameters) and

a direction to a neighboring vertex (1 parameter). It can be shown that the

parameter (or moduli) space P of all icosahedra is a 3-manifold diﬀeomor-

phic to Σ(2, 3, 5) and that it has the same homology groups as the sphere

. Poincar´e showed that π

(P ), the fundamental group of P , is non-trivial.

Since π

)istrivial,P cannot be homeomorphic to S

. Yet another de-

scription of P is obtained by observing that the rotation group SO(3) maps

to itself and the induced action on P is transitive. The isotropy group I

of a ﬁxed point x ∈ P can be shown to be a ﬁnite group of order 60. Thus, P

is homeomorphic to the coset space SO(3)/I

. This fact can be used to show

that π

(P ) is a perfect group of order 120.

Icosahedron is one of the ﬁve regular polyhedra or solids known since

antiquity. They are commonly referred to as Platonic solids (see Appendix

B for some interesting properties of Platonic solids). Poincar´e’s counterexam-

ple showed that homology was not enough to characterize S

and that one

has to take into account the fundamental group. He then made the following

conjecture:

Let M be a closed simply connected 3-manifold. Then M is homeomor-

phic to S

We give it in an alternative form, which is usrful for stating the generalized

Poincar´e conjecture in any dimension, with 3 replaced by a natural number n.

Poincar´eConjecture: Let M be a closed connected 3-manifold with the

same homotopy type as the sphere S

.ThenM is homeomorphic to S

Deﬁnition 2.2 Let H

denote the n-dimensional hyperbolic space. A 3-

manifold X is said to be a Thurston 3-dimensional geometry if it is

one of the following eight homogeneous manifolds (i.e., the group of isome-

tries of X acts transitively on X).

• Three spaces of constant curvature R

• Two product spaces R × S

, R ×H

• Three twisted product spaces, each of which is a Lie group with a left

invariant metric. They are

◦ the universal covering space of the group SL(2, R);

◦ the group of upper triangular matrices in M

(R) with all diagonal en-

tries 1,calledNil;

◦ the semidirect product of R and R

,whereR acts on R

through mul-

tiplication by the diagonal matrix diag(t, t

−1

),t∈ R,calledSol.

64 2 Topology

We can now deﬁne a geometric 3-manifold in the sense of Thurston.

Deﬁnition 2.3 A 3-manifold M is said to be geometric if it is diﬀeo-

morphic to X/Γ ,whereΓ is a discrete group of isometries of X (i.e.

Γ<Isom(X)) acting freely on X,andX is one of Thurston’s eight 3-

dimensional geometries.

We can now state the Thurston geometrization conjecture.

Thurston Geometrization Conjecture::LetM be a closed 3-manifold

that does not contain two-sided projective planes. Then M admits a con-

nected sum decomposition and a decomposition along disjoint incompress-

ible tori and Klein bottles into a ﬁnite number of pieces each of which is a

geometric manifold.

The Thurston geometrization conjecture was recently proved by Perelman

using a generalized form of Hamilton’s Ricci ﬂow technique. This result im-

plies the original Poincar´e conjecture. We will comment on it in Chapter 6.

2.6.3 Topology of 4-manifolds

The importance of the intersection form for the study of 4-manifolds was al-

ready known since 1940 from the following theorem of Whitehead (see Milnor

and Husemoller [285]).

Theorem 2.9 Two closed, 1-connected, 4-manifolds are homotopy equiva-

lent if and only if their intersection forms are equivalent.

In the category of topological manifolds a complete classiﬁcation of closed,

1-connected, oriented 4-manifolds has since been carried out by Freedman

(see [136]). To state his results we begin by recalling the general scheme of

classiﬁcation of symmetric, non-degenerate, unimodular, bilinear forms (re-

ferred to simply as “forms” in the rest of this section) on lattices (see Milnor

and Husemoller [285]). The classiﬁcation of forms has a long history and is an

important area of classical mathematics with applications to algebra, num-

ber theory, and more recently to topology and geometry. We have already

deﬁned two fundamental invariants of a form, namely its rank b

and its

signature (b

−

). We note that sometimes the signature is deﬁned to be

the integer b

− b

−

= b

− 2b

−

. We shall denote this integer by σ(M), i.e.,

σ(M):=b

−2b

−

. We say that a form ι on the lattice L is even or of type II

if ι(a, a) is even for all a ∈ L. Otherwise we say that it is odd or of type I.It

can be shown that for even (type II) forms, 8 dvides the signature σ(M). In

particular, 8 divides the rank of a positive deﬁnite even form. The indeﬁnite

forms are completely classiﬁed by the rank, signature, and type. We have the

following result.

Theorem 2.10 Let ι be an indeﬁnite form of rank r and signature (j, k),j>

0,k > 0. Then we have

2.6 Topological Manifolds 65

1. ι

∼

j(1) ⊕k(−1) if it is odd (type I),

2. ι

∼

mσ

⊕ pE

,m>0ifitiseven(typeII),

where (1) and (−1) are 1 × 1 matrices representing the two possible forms

of rank 1, σ

is the Pauli spin matrix deﬁned earlier, and E

is the matrix

associated to the exceptional Lie group E

in Cartan’s classiﬁcation of simple

Lie groups, i.e.,

⎛

⎜

⎝

2 −1000000

−12−100000

0 −12−10000

00−12−1000

000−12−10−1

0000−12−10

00000−12 0

0000−1002

⎞

⎟

⎠

The classiﬁcation of deﬁnite forms is much more involved. The number

N(r) of equivalence classes of deﬁnite forms (which counts the inequivalent

forms) grows very rapidly with the rank r of the form, as Table 2.3 illustrates.

Tab le 2. 3 Number of inequivalent deﬁnite forms

r 8 16 24 32 40

N(r) 1 2 24 ≥ 10

≥ 10

We now give some simple examples of computation of intersection forms

that we will use later.

Example 2.8 We denote H

(M; Z) by L in this example.

1. Let M = S

;thenL =0and hence ι

= ∅.

2. Let M = S

× S

;thenL has a basis of cohomology classes α, β dual to

the homology cycles represented by S

×{(1, 0, 0)} and {(1, 0, 0)}×S

respectively. With respect to this basis the matrix of ι

is the Pauli spin

matrix





3. Let M = CP

;thenL = Z and hence ι

=(1).

4. Let M =

, i.e., CP

with the opposite complex structure and orien-

tation. Then L = Z and hence ι

=(−1).

Whitehead’s Theorem 2.9 stated above says that the map ι that associates

to a closed, 1-connected topological 4-manifold its intersection form induces

66 2 Topology

an injection from the homotopy equivalence classes of manifolds into the

equivalence classes of forms. It is natural to study this map ι in greater

detail. One can ask, for example, the following two questions.

1. Is the map ι surjective? I.e., given an intersection form μ,doesthereexist

a manifold M such that ι

= μ?

2. Does the injection from the homotopy equivalence classes of manifolds

into the equivalence classes of forms extend to other equivalence classes of

manifolds?

The ﬁrst question is an existence question while the second is a uniqueness

question. These questions can be restricted to diﬀerent categories of manifolds

such as topological or smooth manifolds. A complete answer to these ques-

tions in the topological category is given by the following theorem, proved

by Freedman in 1981 [136].

Theorem 2.11 Let M

(resp., M

) denote the set of topological equiva-

lence classes (i.e. homeomorphism classes) of closed, 1-connected, oriented,

spin (resp., non-spin) 4-manifolds. Let I

(resp. I

) denote the set of equiv-

alence classes of even (resp. odd) forms. Then we have the following:

1. the map ι : M

→I

is bijective;

2. the map ι : M

→I

is surjective and is exactly two-to-one. The two

classes in the preimage of a given form are distinguished by a cohomology

class κ(M) ∈ H

(M; Z

) called the Kirby–Siebenmann invariant.

We note that the Kirby–Siebenmann invariant represents the obstruc-

tion to the existence of a piecewise linear structure on a topological manifold

of dimension ≥ 5. Applying the above theorem to the empty rank zero form

provides a proof of the Poincar´e conjecture for dimension 4. Freedman’s the-

orem is regarded as one of the fundamental results of modern topology.

In the smooth category the situation is much more complicated. It is well

known that the map ι is not surjective in this case. In fact, we have the

following theorem:

Theorem 2.12 (Rochlin) Let M be a smooth, closed, 1-connected, oriented,

spin manifold of dimension 4.Thenσ(M), the signature of M is divisible by

16.

Now, as we observed earlier, 8 always divides the signature of an even

form, but 16 need not divide the form. Thus, we can deﬁne the Rochlin

invariant ρ(μ)ofanevenformμ by

ρ(μ):=

σ(μ)(mod 2).

We note that the Rochlin invariant and the Kirby–Siebenmann invariant

are equal in this case, but for non-spin manifolds the Kirby–Siebenmann

invariant is not related to the intersection form and thus provides a further

obstruction to smoothability. From Freedman’s classiﬁcation and Rochlin’s

2.6 Topological Manifolds 67

theorem it follows that a topological manifold with nonzero Rochlin invariant

is not smoothable. For example, the topological manifold |E

| := ι

−1

)

corresponds to the equivalence class of the form E

and has signature 8

(Rochlin invariant 1) and hence is not smoothable.

For several years very little progress was made beyond the result of the

above theorem in the smooth category. Then, in 1982, through his study of

the topology and geometry of the moduli space of instantons on 4-manifolds

Donaldson discovered the following, unexpected, result. The theorem has led

to a number of important results including the existence of uncountably many

exotic diﬀerentiable structures on the standard Euclidean topological space

Theorem 2.13 (Donaldson) Let M beasmoothclosed1-connected oriented

manifold of dimension 4 with positive deﬁnite intersection form ι

.Then

∼

(1), the diagonal form of rank b

, the second Betti number of M.

Donaldson’s work uses in an essential way the solution space of the Yang–

Mills ﬁeld equations for SU(2) gauge theories and has already had profound

inﬂuence on the applications of physical theories to mathematical problems.

In 1990 Donaldson obtained more invariants of 4-manifolds by using the

topology of the moduli space of instantons. Donaldson theory led to a number

of new results for the topology of 4-manifolds, but it was technically a diﬃcult

theory to work with. In fact, Atiyah announced Donaldson’s new results

at a conference at Duke University in 1987, but checking all the technical

details delayed the publication of his paper until 1990. The matters simpliﬁed

greatly when the Seiberg–Witten equations appeared in 1994. We discuss

the Donaldson invariants of 4-manifolds in more detail in Chapter 9. It is

reasonable to say that at that time a new branch of mathematics which may

be called “Physical Mathematics” was created.

In spite of these impressive new developments, there is at present no ana-

logue of the geometrization conjecture in the case of 4-manifolds. Here ge-

ometric topologists are studying the variational problems on the space of

metrics on a closed oriented 4-manifold M for one of the classical curvature

functionals such as the square of the L

norm of the Riemann curvature Rm,

Weyl conformal curvature W , and its self-dual and anti-dual parts W

and

−

, respectively, and Ric, the Ricci curvature. The Hilbert–Einstein varia-

tional principle based on the scalar curvature functional and its variants are

important in the study of gravitational ﬁeld equations. Einstein metrics, i.e.,

metrics satisfying the equation

K := Ric −

Rg =0

are critical points of all of the functionals listed above. Here K is the trace-free

part of the Ricci tensor. In many cases the Einstein metrics are minimizers,

but there are large classes of minimizers that are not Einstein metrics. A well-

known obstruction to the existence of Einstein metrics is the Hitchin–Thorpe